Conclusion and Outlook

Abstract

An obvious question for the conclusion may be: What did the reader learn from this book? At the core of it there were two approaches to pulse-coupled neural nets: the lighthouse model and integrate and fire models. I tried a systematic exposition, which made all calculations explicit. So I am sure the reader could follow step by step. Let us first recapitulate our results on the lighthouse model. We established the phase-locked state, whereby an important issue is the stability of that state when various parameters are changed. Such parameters are the external (sensory) signals, the synaptic strengths, which may lead to excitatory or inhibitory interactions, and the various delay times. The lighthouse model had the advantage that all the calculations could be done rather easily. In this way, we could discuss the stability limits and also the effect of pulse shortening. The phase-locked state was established by means of a specific initial condition that does not seem unrealistic. We then studied the kind of stability, whether marginal or asymptotic. We recognized that this feature depends on the kind of perturbations, where we studied those that lead only to marginal stability as well as those that lead to asymptotic stability. The integrate and fire models, of which we presented several versions, are mathematically more involved. We focussed our attention on the phaselocked or synchronized state that could be evaluated for all the versions presented.